How to split a Box or a Mesh with a Plane and keep both sides?
This can be done using the Bool Tool add on.  Bool Tool is included with Blender.  It needs enabling.
Step 1 Enable Bool Tool: 
Step 2:  I have a Cube that i have intersected with a plane.

Step 3: Select the Plane first, hold down Shift key then Select the Cube

Step 4:  From the Properties Panel (Press N if its not visible)  go to Edit then Bool Tool

Step 5:  Use the Slice option in Either the Auto Boolean or Brush Boolean menus.
Result is a Cube cut into two sections with faces applied between the two.